Pila on a “modular Fermat equation”

I like this paper by Pila that just went up on the arXiv, which shows the way that you can get Diophantine consequences from the rapid progress being made in theorems of Andre-Oort type.  (I also want to blog about Tsimerman + Zhang + Yuan on “average Colmez” and Andre-Oort, maybe later!)

Pila shows that if N and M are sufficiently large primes, you can’t have elliptic curves E_1/Q and E_2/Q such that E_1 has an N-isogenous curve E_1 -> E’_1, E_2 has an M-isogenous curve E_2 -> E’_2, and j(E’_1) + j(E’_2) = 1.  (It seems to me the proof uses little about this particular algebraic relation and would work just as well for any f(j(E’_1),j(E’_2)) whose vanishing didn’t cut out a modular curve in X(1) x X(1).)  (This is “Fermat-like” in that it asserts finiteness of rational points on a natural countable family of high-genus curves; a more precise analogy is explained in the paper.)

How this works, loosely:  suppose you have such an (E_1, E_2).  A theorem of Kühne guarantees that E_1 and E_2 are not both CM (I didn’t know this!) It follows (WLOG assume N > M) that the N-isogenies of E_1 are defined over a field of degree at least N^a for some small a (Pila uses more precise bounds coming from a recent paper of Najman.)  So the Galois conjugates of (E’_1, E’_2) give you a whole bunch of algebraic points (E”_1, E”_2) with j(E”_1) + j(E”_2) = 1.

So what?  Rational curves have lots of low-height algebraic points.  But here’s the thing.  These isogenous choices of (E’_1, E’_2) aren’t just any algebraic points on X(1) x X(1); they represent pairs of elliptic curves drawn from a {\em fixed pair of isogeny classes}.  Let H be the hyperbolic plane as usual, and write (z,w) for a point on H x H corresponding to (E’_1, E’_2).  Then the other choices (E”_1, E”_2) correspond to points (gz,hw) with g,h in GL(Q).  GL(Q), not GL(R)!  That’s what we get from working in a fixed isogeny class.  And these points satisfy

j(gz) + j(hw) = 1.

To sum up:  you have a whole bunch of rational points (g,h) on GL_2 x GL_2.  These points are pretty low height (for this Pila gestures at a paper of his with Habegger.)  And they lie on the surface j(gz) + j(hw) = 1.  But this surface is a totally non-algebraic thing, because remember, j is a transcendental function on H!  So (Pila’s version of) the Ax-Lindemann theorem (correction from comments:  the Pila-Wilkie theorem) generates a contradiction; a transcendental curve can’t have too many low-height rational points.

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One thought on “Pila on a “modular Fermat equation”

  1. Jacob says:

    Nice post!

    I wanted to clarify a slight error: it is not the Ax-Lindemann theorem that yields a contradiction at the end. Its the Pila-Wilkie theorem that shows that transcendental things don’t have too many low-height rational points.

    Ax-Lindemann is what shows that algebraic equations in j don’t contain any `actual’ algebraic things, without a hecke-curve lying somewhere in there.

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